H ∞ Tracking Control of Non-lower Triangular Nonlinear Systems: Coronary Artery System

. This work presents a novel control approach that integrates backstepping and dynamic surface control techniques to design an H ∞ tracking control system. Our focus is on a particular class of nonlinear systems that do not possess lower triangular structure, with a specific application to the coronary sys-tem. By employing dynamic surface control, the issue of di ff erential explosion encountered during the backstepping process is e ff ectively addressed, while also guaranteeing robustness. Finally, simulation of coronary artery system proves the e ff ectiveness of the proposed control strategy.


Introduction
The coronary artery system (CAS) plays a pivotal role in maintaining stable heart function, as any sudden pressure variations or occlusions can give rise to arrhythmias, characterized by chaotic blood vessel behavior, resulting in fast, slow, or irregular heart rhythms.This intricate system is subject to fluctuations in inner pressure and diameter, influenced by various unpredictable factors such as heart ectopic pacemakers, thyroid disease, high blood pressure, dynamic irregularities, and heart valve issues.The integrity of a healthy coronary artery vessel is paramount in restraining arrhythmias; failure to do so can culminate in cardiac arrest and even mortality.Given the profound impact of this system on overall health, much research has been devoted to it.
In 1986, [1] introduced a mathematical model for the coronary artery system along with key parameter values, establishing mathematically that coronary artery spasm embodies a nonlinear chaotic system.Acknowledging the susceptibility of chaotic systems to initial conditions or disturbances, which can potentially result in catastrophic chaos within a coronary artery, controlling this system becomes imperative to prevent myocardial infarction [2].As shown in [3] and [4], the ability to employ minor disturbances to control the cardiac chaos and restore normal cardiac function when deviations occur holds promise to avoid coronary artery spasm.
The field of non-linear control has garnered significant attention, leading to the development of numerous control strategies aimed at tackling non-linear dynamics.These approaches encompass adaptive control methods [5,6], sliding mode control techniques [7,8], and T-S fuzzy control strategies [9,10].Additionally, the backstepping method, as introduced by Kokotovic [11], has been employed to address the intricate control challenges posed by nonlinear systems.A plethora of results have emerged from this research endeavor, as documented in [12][13][14][15][16][17][18][19][20].Nonetheless, it is crucial to recognize two inherent limitations associated with the backstepping control method.Firstly, the complexity of control design and stability analysis undergoes exponential growth when applied to nonlinear systems of lower non-triangular form, a phenomenon commonly called "complexity explosion" in this context.Secondly, backstepping control exhibits sensitivity to measurement noise, which can adversely affect its performance for interconnected systems.
On the side of complexity explosion problem, Swaroop et al. [21] introduced a dynamic surface control (DSC) technique.This approach integrated a low-pass filter was in each control input, which has the advantage of zeroing the derivatives of the virtual control inputs.Thus, it eliminates expansions of differential elements, and consequently reduces controller complexity.On the other side, H ∞ robust control stands as an effective method to achieve stabilization with guaranteed performance.In this context, [22], studied H ∞ control problem to synchronize CAS with it nominal system subjected to disturbances and input delay.Li et al. [23] investigated H ∞ control for CAS via free-matrix-based integral inequality with time-delay.The authors in [24] considered the CAS for H ∞ synchronization problems with input time-varying delay and input disturbances.
Given the CAS imperative need for a control approach capable of surmounting the complexity explosion challenge while concurrently demonstrating robustness, the motivation behind this paper lies in the development of a robust H ∞ tracking control synergized with dynamic surface control.
To address the limitations associated with backstepping, including complexity explosion and sensitivity to measurement errors (Section 2), this paper presents a novel robust H ∞ tracking controller design approach for systems characterized by lower non-triangular form (Section 3), specifically focusing on the coronary artery system.This method combines dynamic surface and backstepping control techniques.Compared to existing findings, the proposed controller scheme effectively eliminates the need for repeated differentiation of virtual controls, all while establishing a robust H ∞ performance metric.Illustrative simulations are provided for the coronary artery system (Section 4).Finally, we present a conclusion with the research line for expected future works (Section 5).

System description
Consider the general class of the coronary artery system defined by, where of the coronary artery system adopted in [25], by specifying x ∈ R 2 , with x 1 and x 2 are the inner diameter changes and the pressure of coronary artery vessel of CAS, respectively; the control input u(t) is the potency or dosage of the single and double nitrate isosorbide nitrate (isosorbide mononitrate and dinitrate) or the nitroglycerin used for treating angina and other diseases; ∆ 1 stands for the uncertainty from the vessel in radius; and ∆ 2 describes the uncertainty from the pressure change of the vessel.Figure 1 illustrates the healthy CAS and disease CAS.Additional parameters are given in Table 1.

Objectives
The medical objective is oriented towards achieving synchronization between the states of a Coronary Artery Disease (CAD) system and those of a healthy vessel.To this end, the primary aim of the present study is to formulate a controller design such that the output y (pertaining to the CAD system) effectively tracks a reference signal y r (generated by the dynamics of a normal CAS ).This pursuit also entails achieving an H ∞ performance index γ, which quantifies the extent of disturbance attenuation.The formal definition of the control objective is as follows: Definition 1 [26] The H ∞ tracking problem of system (1) is said to be solvable if there exists a state feedback control law u so that the closed-loop system satisfies the following properties: 1.The tracking error s 1 (t) = y(t) − y r (t) of the resulting closed-loop system (1) is asymptotically stable with ∆ = 0; where γ > 0 is the disturbance attenuation rate and

Problem definition
Given the preformance index J in (3), construct a control law u for the dynamic model ( 1) such as all closed-loop signals are bounded and the output tracking errors converges towards a small neighborhood of zero for all t ∈ R + .

Backstepping method and non-lower triangular nonlinear systems
Why standard back-stepping control approach is not working for CAS?
The conventional back-stepping control methodology encounters limitations when employed for the CAS system defined by Equation (1).In this context, the conventional back-stepping strategy fails to achieve desired outcomes due to the absence of a lower triangular structure within the system, thus impairing its efficacy.Upon employing the standard back-stepping approach, a virtual control law α i+1 is derived in each intermediate step (i = 1, 2, . . ., n) to stabilize the i th subsystem.However, complications arise due to the inclusion of the state variable x n within α i+1 , leading to the incorporation of the actual control input u within its derivative.When α i+1 is directly injected into the x i+1 subsystem, the subsequent virtual control law α i+2 becomes inherently contingent on the forthcoming real control input u.This interdependence engenders a cycle of complexity in control law relationships, commonly termed "complexity explosion."Illustratively, consider a coronary system scenario wherein a pivotal control objective is to facilitate convergence of the tracking error s 1 = x 1 − y r toward a vicinity of zero.Employing the standard back-stepping methodology, the initial step involves evaluating the time derivative of s 1 as expressed below: Suppose we were to design x 2 as −k 1 s 1 − f 1 (x 1 , x 2 ) + ẏr .In that case, subsystem (4) could be transformed into ṡ1 (t) = −k 1 s 1 , consequently ensuring asymptotic convergence of the tracking error s 1 to zero.However, as x 2 constitutes a system state variable that cannot be tailored arbitrarily, introducing a virtual control law α 2 = −k 1 s 1 − f 1 (x 1 , x 2 ) + ẏr becomes imperative.The crux of the design objective is to facilitate the convergence of x 2 − α 2 to zero.Notably, the presence of x 1 and x 2 within α 2 implies that the derivatives of the nonlinear functions f 1 (x 1 , x 2 ) will manifest in the formulation of α 2 .Consequently, this intricate interplay leads to the intricate design of the actual control input u if the direct injection of α 2 into the subsequent subsystem ( ṡ2 (t) = ẋ2 − α2 ) is pursued.

Proposed control approach:
In order to effectively mitigate the challenges posed by the aforementioned complexity explosion, the present study employs a novel approach rooted in the (DSC) technique for control system design.This method entails the utilization of a first-order filter to modulate the virtual control signal α i+1 , resulting in the generation of an output signal ω(t) from the filter.Subsequently, this filtered signal ω(t) is introduced into the subsequent subsystem.In the culminating stage of this control methodology, specifically the final step denoted as step n, a tangible control law u(t) is meticulously devised to achieve the stabilization of the n subsystems.
Without loss of generality, the proposed approach relies on following technical assumptions.
Assumption 1 For every i ∈ 1, . . ., n, the functions f i and g i are bounded and n − i times continuously differentiable.

Assumption 3
The desired trajectory y r is a known and bounded function with time, and its derivatives are also known and bounded.

H ∞ robust controller design
In this section, a backstepping based DSC design scheme is established for the general form of coronary artery system (1), will be devised.Firstly, define the desired trajectory as y r (t), so the output tracking error is s 1 (t) = y(t) − y r (t).Introducing the following coordinate transformation: where ω i (t) represents the output signal of the low-pass filter with α i (t) as the input signal, and meets: where τ i is a filtering parameter.The approximation error of above filters are determined by The implementation of the control algorithm is provided by the following Theorem 1.

E3S Web of
and actual control law satisfies: then, the system described in (1) is asymptotically stable and meets the H ∞ performance.All signals are bounded in the closed loop system, and output tracking error can converge to a small neighborhood.
Next, the sufficiency of the Theorem 1 is proved in the following, and the whole process of proof includes n steps.

Proof 1
Step 1: With the aid of s 1 = x 1 − y r , its derivative is given by Choose the following Lyapunov function: Define the following function: Substituting the derivative of V 1 (s 1 ) , x 2 = s 2 + ω 2 , and ω 2 = ρ 2 + α 2 into (12), we obtain where Γ 1 = 1/γ 2 + 1/2, and the inequality above is obtained by using the following relation: Introduce the virtual control α 2 as follows: where k 1 is a positive design parameter.It follows from substituting α 2 into (13) that Step 2: It is easily obtained that Select a Lyapunov function candidate: By differentiating both sides of (18), the following relation can be obtained: Substituting (12) into the equation above gives Define the following function: Then, by substituting (20) into (21) and taking (16) into consideration, it follows that where where the inequality is obtained by using the following relation: E3S Web of Conferences 469, 00074 (2023) ICEGC'2023 https://doi.org/10.1051/e3sconf/202346900074 Define the virtual control α 3 as follows: where k 1 is a positive design parameter.It follows from substituting α 3 into (23) that Step i i(i = 3, . . ., n − 1): Suppose that at Step i − 1, the function satisfies the inequality where Consider the i th subsystem The following Lyapunov function candidate: By differentiating both sides of (31), the following can be obtained: Substituting (29) and (27) into the equation above yields Define the function Then, by using (33) and (28), it can be easily verified that Substituting x i+1 = s i+1 + ω i+1 and ω i+1 = ρ i+1 + α i+1 into (35) gives where the inequality is obtained by Design the virtual control α i+1 as follows: where k i is a positive design parameter.It follows from substituting α i+1 into (36) that Step n: Repeating step similar to step i, the following equation is satisfied Select Lyapunov function candidate as follows The following relation can be obtained by differentiating both sides of (39) and using (41) with i = n − 1 : Vn (s 1 , . . ., s n ) = Vn−1 (s 1 , . . ., s n−1 ) + s n ṡn (42) Define the following function: Choose the control input u as follows: where k i is a positive design parameter.It follows from substituting u into (43) that Select V N (s 1 , . . ., s n ) = 2V n (s 1 , . . ., s n ).Then it follows from (43) that the derivative of V N satisfies Because of P n ≤ 0, the following inequality is obtained: By integrating both sides of inequality (48), inequality (3) in Definition 1 can be obtained, which indicates that the L 2 gain from uncertainties ∆i to output s 1 is smaller than or equal to a positive constant γ.
So far, the tracking control design has been accomplished based on the backsteping algorithm.
The block diagram of the design procedure for the proposed controller given in section 3 is shown in Figure 4.The simulation is executed by using the design parameters above, and all the simulation response curves are illustrated in figures 4 -8.As you can see from figure 4, the system output y(t) follow the reference signal y r (t) quickly and precisely.figure 5 illustrates the tracking error trajectory.Obviously, it is found that the tracking error can converge to a small neighborhood of zero.The response curves of the states x 1 , x 2 and the control law u are illustrated in figure 6 and figure 7, respectively, which illustrate distinctly that all the signals are bounded stable.Figure 8 shows the trajectories of the input and output signals of the first-order low-pass filters.Therefore, it is easy to see that the proposed method can achieve a better performance with small control gains, and the proposed control design scheme is feasible and effective.

Conclusion
This work formulated a backstepping control strategy tailored to address the complexity explosion and robustness challenges in coronary artery system.Leveraging Dynamic Surface Control and H ∞ guaranteed performance, we have effectively circumvented the need for virtual control differentiation at each recursive step during the backstepping design process while guaranteeing robustness arising from disturbances.The simulation results showed significant enhancements in tracking performance.
For future endeavors, it is worth exploring the integration of neural or fuzzy adaptive control methodologies.These approaches can be instrumental in approximating uncertain terms within the CAS system, further enhancing the control precision and adaptability of our framework.

Figure 1 :
Figure 1: Presentation of healthy CAS and disease CAS

Figure 2 :
Figure 2: A simplified schematic illustration of the study.

E3SFigure 3 :
Figure 3: The block diagram of the design procedure for the proposed controller.

Figure 4 :
Figure 4: Trajectroies of y and y r of the proposed method.

Figure 5 :Figure 6 :
Figure 5: Response curve of the tracking error s 1 .

Table 1 :
[25]re the system states; u(t) ∈ R and y(t) ∈ R represent the system input and output signals, respectively; f i (.) is known nonlinear function; g n (•) is a continuous function and satisfies g n (•) 0; ∆ i is bounded disturbances.It is possible to recover the case System parameters[25] Given positive scalars γ, Γ 1 , k i , with Γ 1 = 1/γ 2 + 1/2, if the following virtual control laws satisfy: s 1 , . . ., s n ) .